January16 ,2019 Part2:Antiderivativesandindefiniteintegrals Calculus!

MAT137Y1 – LEC0501

Calculus!

Part 2:

Antiderivatives and indefinite integrals

January 16 ^th , 2019

Initial Value Problem

Find a function 𝑓 ∶ ℝ → ℝ such that

• For every 𝑥 ∈ ℝ, 𝑓 ^″ (𝑥) = sin 𝑥 + 𝑥 ² ,

• 𝑓 ^′ (0) = 5,

• 𝑓 (0) = 7.

Towards FTC

−1 1 2 3 4 5

−2

−1 1 2

𝑂

𝑦 = 𝑓 (𝑥)

Compute:

1 ∫

1 0

𝑓 (𝑡)𝑑𝑡

2 ∫

2 0

𝑓 (𝑡)𝑑𝑡

3 ∫

3 0

𝑓 (𝑡)𝑑𝑡

4 ∫

4 0

𝑓 (𝑡)𝑑𝑡

5 ∫

5 0

𝑓 (𝑡)𝑑𝑡

Towards FTC (continued)

−1 1 2 3 4 5

−2

−1 1 2

𝑂

𝑦 = 𝑓 (𝑥)

Call 𝐹 (𝑥) =

∫

𝑥 0

𝑓 (𝑡)𝑑𝑡. This is a new function.

• Sketch the graph of 𝑦 = 𝐹 (𝑥).

• Using the graph you just sketched, sketch the graph of

Trig-exp antiderivatives

1 Compute 𝑑

𝑑𝑥 [ 𝑒 ^𝑥 sin 𝑥] , 𝑑

𝑑𝑥 [ 𝑒 ^𝑥 cos 𝑥] .

2 Use the previous answer to Compute

∫ 𝑒 ^𝑥 sin 𝑥 𝑑𝑥,

∫ 𝑒 ^𝑥 cos 𝑥 𝑑𝑥.

A harder antiderivative

1 Compute 𝑑

𝑑𝑥 [arctan 𝑥] , 𝑑 𝑑𝑥 [

𝑥 1 + 𝑥 ² ] .

2 Use the previous answer to compute

∫

1 (1 + 𝑥 ² ) ²

𝑑𝑥